{"id":156133,"date":"1999-08-01T00:00:00","date_gmt":"1999-08-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/logic-with-equality-partisan-corroboration-and-shifted-pairing\/"},"modified":"2018-10-16T20:15:06","modified_gmt":"2018-10-17T03:15:06","slug":"126-logic-with-equality-partisan-corroboration-and-shifted-pairing","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/126-logic-with-equality-partisan-corroboration-and-shifted-pairing\/","title":{"rendered":"Logic with Equality: Partisan Corroboration and Shifted Pairing"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Herbrand&#8217;s theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a quantifier free formula, one seeks a valid disjunction of m instantiations of that formula. In the presence of equality, which is the case in this paper, this problem was recently shown to be undecidable. The main contributions of this paper are two theorems. The first, the Partisan Corroboration Theorem, relates corroboration problems with different multiplicities. The second, the Shifted Pairing Theorem, is a finite tree automata formalization of a technique for proving undecidability results through direct encodings of valid Turing machine computations. These theorems are used in the paper to explain and sharpen several recent undecidability results related to the corroboration problem, the simultaneous rigid E-unification problem and the prenex fragment of intuitionistic logic with equality.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Herbrand&#8217;s theorem plays a fundamental role in automated theorem proving methods based on tableaux. The crucial step in procedures based on such methods can be described as the corroboration problem or the Herbrand skeleton problem, where, given a positive integer m, called multiplicity, and a quantifier free formula, one seeks a valid disjunction of m 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